Step 1

Whisk lime juice, fish sauce, vinegar, brown sugar, and garlic in a small bowl until sugar is dissolved; set dressing aside.Do ahead: Dressing can be made 1 day ahead. Cover and chill.

Step 2

Bring farro, 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and 1 quart water to a boil in a medium pot. Reduce heat, cover, and simmer until farro is tender but still chewy, 20–25 minutes. Drain off any excess liquid and let cool.

Step 3

Prepare a grill for medium-high heat. Grill corn, turning occasionally, until kernels are charred, 10–15 minutes. Transfer to a cutting board and let sit until cool enough to handle. Cut kernels from cobs.

Step 4

Combine farro, corn, plums, chiles, mint, and ½ cup Thai basil in a large bowl and drizzle reserved dressing over; toss to coat. Taste and season salad with more salt if needed. Transfer to a platter and top with more Thai basil leaves.Do ahead: Farro can be cooked 1 day ahead. Transfer to an airtight container; cover and chill.